Soldiers of Fortune goes to Hollywood

There are six identical men, who must choose one of them to do an unpleasant job. They could hold an auction and pay one of them to volunteer to do the job. But if they have diminishing marginal utility of consumption, they will prefer instead to roll a die to decide which one of them does the job. Their expected utility is higher. If the job is unpleasant because it has a risk of death, then state-contingent preferences (the marginal utility of consumption when dead is zero) strengthen the argument for a lottery.

That's the intuition behind a paper I read 25 years ago, about lotteries and the draft for the Confederate Army. I think the title was "Soldiers of Fortune", but I don't remember the author, and I can't find it on the internet. Sorry. by Ted Bergstrom.

Now imagine that a seventh man controls the die, and can offer deals to the other six. He can collect rents in exchange for tilting the die. But there is a limit to how much rent he can collect, or the six will instead auction the job among themselves.

Now change the model, reversing the sign, so the job that one of them will do becomes pleasant. One of them will become a movie star. With diminishing marginal utility of consumption, they will prefer to roll a die to decide who gets to be the star. And with state-contingent preferences (being a star increases the marginal utility of consumption) that strengthens the argument for a lottery.

And again, if a seventh man controls the die, he can collect rents. But there's a difference, because in this case the seventh man need only make a deal with one of the six, not with five of the six. If the other five don't know about the secret deal, they won't revert to the auction.

[Trying to answer a question raised by Steve Randy Waldman's tweet.]

Comments

[Tel]

If you have six identical men, then by definition you cannot have an economy because that requires division of labour and hence non-identical men.

Therefore none of them owe anything to the others, and have no particular reason to do the job, with or without a roll of the die.

Besides that, the Confederate Army lost the war, so everyone would have been better off if they refused to fight (including but not limited to themselves).

[Nick Rowe]

Tel: Division of Labour can also be based on Economies of Scale (Adam Smith), as well as Comparative Advantage (David Ricardo).

[Tel]

"First, the improvement of the dexterity of the workmen, necessarily increases the quantity of the work he can perform; and the division of labour, by reducing every man's business to some one simple operation, and by making this operation the sole employment of his life, necessarily increases very much the dexterity of the workman. A common smith, who, though accustomed to handle the hammer, has never been used to make nails, if, upon some particular occasion, he is obliged to attempt it, will scarce, I am assured, be able to make above two or three hundred nails in a day, and those, too, very bad ones. A smith who has been accustomed to make nails, but whose sole or principal business has not been that of a nailer, can seldom, with his utmost diligence, make more than eight hundred or a thousand nails in a day. I have seen several boys, under twenty years of age, who had never exercised any other trade but that of making nails, and who, when they exerted themselves, could make, each of them, upwards of two thousand three hundred nails in a day. The making of a nail, however, is by no means one of the simplest operations. The same person blows the bellows, stirs or mends the fire as there is occasion, heats the iron, and forges every part of the nail: in forging the head, too, he is obliged to change his tools. The different operations into which the making of a pin, or of a metal button, is subdivided, are all of them much more simple, and the dexterity of the person, of whose life it has been the sole business to perform them, is usually much greater. The rapidity with which some of the operations of those manufactures are performed, exceeds what the human hand could, by those who had never seen them, be supposed capable of acquiring."

When I read that, it sounds to me very much like a description of individual specialization. Even if you believe that people start out identical, you won't get a non-linear economy of scale unless those people start to focus on particular tasks and improve their skills within a narrow area. Emphasis has been added by me, not in the original.

[Nick Rowe]

Tel: simplest example: the 6 men only learn 1 job each, instead of each learning all 6 jobs. Specialisation saves on training costs.

[louis]

How does bargaining go between the keeper of the die and the person with whom he cuts the deal?
Keeper to person 1: I will guarantee you win, improving your expected value 6-fold, if you pay me a 90% of the increase in EV
Person 1: I'll give you 1% of the benefit.
Keeper: I'll go to Person 2 and get a better deal
Person 1: I'll tell 3-6 and they'll find a new keeper or revert to auction
....?

[Tel]

"Specialisation saves on training costs."

Yup, and it also make people non-identical.

Hence a Plumber is not identical to an Electrician, is not identical to a Mechanic, is not identical to a Programmer.

[Majromax]

> How does bargaining go between the keeper of the die and the person with whom he cuts the deal?

The Nash equilibrium would be to ask for the difference in EV (less epsilon) between a fair die roll and a fair auction. Perniciously, a deceitful die-keeper could make this deal with each of the six and only follow through once (or roll a fair die, even!) to extract all of the gains from the choice of selection.

[louis]

Majro - when the keeper asks the first of the six for the full difference in EV, the person approached may threaten to defect and tell the other 5 that the keeper is dishonest and should be replaced. Shouldn't this weaken the keeper's ability to extract value?

[Majromax]

If the replacement is with an auction, then doing so is to the first's detriment: they would replace an outcome of (win - small payment to dice-keeper) with [(win - auction price) or (1/5 auction price)].

In fact, writing it out like this changes my thinking. Asking for the difference in EV between an auction and a fair dice roll (which depends on diminishing marginal utility of consumption) is a Nash equilibrium even if the deals are made in public. If the deal is made in private with the expectation of follow-through, the asking price can be much larger but only for a single participant (with honest parties).

The keeper's ability to extract value is only weakened if:

*) The game is repeated, such that more complicated strategies like tit-for-tat are possible, or
*) If the six are altruistic and care about each others' welfare.

For the first, think of this as a simple form without auction, with you as the die-keeper. I can either accept a 1/6 chance of winning $100, or I can pay you to give me $100. Rationally, I should be willing to accept any price up to $83, provided I do not expect the game to be repeated.

[louis]

Majro - very interesting, thanks.
I assume you need consent of all 6 players to authorize the keeper. So at any point pre-roll, if someone objects, they look for another keeper (how I was looking at it) or they default to auction (how you were looking at it).
If the 6 can nominate another keeper, that definitely weakens the current keeper's hand. Once a keeper reveals that he is crooked by seeking to strike a deal with one party, that party can threaten to use that info to take away the keeper's ability to extract rent, by disclosing his crookedness to the others or simply objecting himself.
But let's say the only alternatives are "Keeper Joe" and auction. If deal is made in public, you need to ensure each of their individual EV's is better than auction EV. So you can tilt the die marginally in favor of up to 5 of the participants, subject to the constraint that the worst off party still needs to be better off than in auction.

[notsneaky]

Comparative advantage requires non-identical agents. Those PPFs need to have different slopes. "Comparative advantages" is just a short hand way of saying "gains from trade arising out of differences"

[notsneaky]

Also "it has a risk of death, ... (the marginal utility of consumption when dead is zero)" is a strange statement from point of view of ordinal utility. Why not -infinity? Or 42?

[notsneaky]

How does this work? The guy who gets paid to do the unpleasant job has to have the same utility as everyone else, right? If it was higher, someone would out bid him. If it was lower, he would never make that bid in the first place. Ok, so:

W is initial wealth, V is cost of doing the unpleasant job and N (greater than 2) is number of individuals. U is the utility function, which is concave.

***With auction each person's utility is U(W-V/(N-1))
***With lottery each person's expected utility is (1/N)*U(W-V)+((N-1)/N)*U(W)

Normalize U(W-V)=0. So we compare U(W-V/(N-1)) vs. ((N-1)/N)*U(W). The question is can I make the left hand side - the auction - bigger than the right hand side, the lottery. If I let V go to zero then LHS goes to U(W) which is of course bigger than ((N-1)/N)*U(W), so yeah, I can do that. Alternatively I could take V as given but choose W and a particular utility function. Say U(x)=x^.5. Then I have (1-V/(W*(N-1))^.5 vs. (N-1)/N. I let W get really large. The LHS goes to 1 and becomes larger.

So it seems like there's lots of cases where the claim is not true. What is missing?

[Nick Rowe]

notsneaky: try it with a utility function like U=U(W)-V, where U(W) is concave. (The job creates disutility, you have set it up so the job takes away some of your wealth, not utility.)

[notsneaky]

So it's about the degree of substitution between wealth and the (lack of) unpleasantness of the job. Which means that if these are perfect substitutes, my comment is still correct?

[Nick Rowe]

notsneaky: yes, I think that's right.